Relativistic Spin-1/2 Fermions

As opposed to the quantum electromagnetic ﬁeld, where corresponding quantum ﬁeld theory was introduced axiomatically, in the case of spin-1/2 particles it is most instructive to start with the single-particle quantum mechanics and then proceed with the standard second-quantization protocol. This way one naturally arrives at two crucial (and deeply related to each other) cir- cumstances. Namely, that (i) the spin-1/2 system should necessarily be a system of fermions1 and (ii) there should necessarily be anti-particles.

Massless particles: Weyl equation

Our goal is to establish the simplest relativistic Hamiltonian for a two- component spinor. The guiding principle is the combination of Lorentz and rotational symmetry allowing one to establish good quantum numbers. In view of translational invariance, the momentum k, i.e., the eigenvalue of the operator −∇, should definitely be a good quantum number, so that it makes a perfect sense to look for the desired Hamiltonian in the momentum representation, where the momentum operator is just a real vector. In non-relativistic physics, spin can be naturally decoupled from momentum, because rotational transformations are totally independent from both translational and Galilean transformations. With Lorentz transformation the sit- uation is radically different. For a spinful field, the Lorentz transformation mixes spin components, so that the only possible spin-related good quantum number in the sector of given k is the so-called helicity, the eigenvalue of the helicity operator (kˆ = k/k) ~σ · kˆ (helicity operator). (1) From the spin-1/2 theory, we know that this operator (of spin projection onto the axis kˆ) has the eigenvalues +1 and −1. The term “helicity” comes by

1We note in passing that this fact is a manifestation of the general spin-statistics theo- rem stating that a consistent relativistic quantum ﬁeld theory implies a strict relationship between the spin and statistics: Half-integer spins correspond to fermions while integer spins correspond to bosons.

1 the rotational analogy. Correspondingly, the +1 state is called right-handed (R), and the −1 state is called left-handed (L). Given that (i) helicity has to be a good quantum number and (ii) the energy has to satisfy the relativistic dispersion law (we seth ¯ = c = 1)

2 2 2 Ek − k = m , (2) for the Hamiltonian Hk (in the sector of momentum k) of massless (m = 0) particles we have only two options:

(±) Hk = ±~σ · k (Weyl Hamiltonians). (3)

In coordinate representation, where

H(±) = ∓i~σ·∇ (Weyl Hamiltonians), (4) corresponding Schr¨odingerequation for a two-component spinor |ψi is called Weyl equation: ∂ i |ψi = H(±) |ψi (Weyl equation). (5) ∂t In both cases, the energy spectrum is really strange. The energies of right- handed and left-handed eigenstates have diﬀerent signs:

(+) (+) Ek,R = k, Ek,L = −k,

(−) (−) Ek,R = −k, Ek,L = k. Most importantly, the existence of the negative energy states renders the theory pathological at the single-particle level. If coupled to the rest of the universe, a single particle becomes an inﬁnite source of energy. The resolution of the paradox was found by Dirac.2 The spin-1/2 relativistic particles have to be fermions. Furthermore, the vacuum state of the system is such that all the negative-energy states are occupied and all the positive-energy states are empty. Given that fermionic creation/annihilation operators are symmetric with respect to the particle-hole transformation,

† † cˆs =a ˆs, cˆs =a ˆs, 2In the context of his theory of relativistic electrons discussed in the next section.

2 the filled negative-energy states are equivalent to the vacuum of holes.3 This way one arrives not only at the fermionic statistics, but also at the necessity of having anti-particles (holes) in the relativistic theory of spin-1/2 quantum fields.4 The problem of negative energies—fixed by fermionic statistics and anti- particles—is not the only problem we face with Weyl equation. Yet another problem is parity violation. Indeed, for each of the two equations, both particles and anti-particles are either right-handed [for the model H(+)], or left-handed [for the model H(−)]. There are two options of how to deal with this circumstance. One option is that the parity violation is the physical property of the system.5 The second option is to double the Hilbert space by combining two spinors into a single four-component bi-spinor  (+)  ψ1    ψ(+)   2  |Φi =  (−)  (6)  ψ1   (−)  ψ2 and, correspondingly, combine the two Hamiltonians into a single 4×4 Hamil- tonian " H(+) 0 # H = . (7) 0 H(−) In the next section, we will see that such a combination is absolutely necessary in a theory with a finite mass.

Massive particles: Dirac equation

To understand the structure of the relativistic Hamiltonian for a massive particle, observe that the relativistic dispersion relation (2) coinsides with the characteristic equation for the matrix " k m # . m −k

3Meaning, in particular, that the holes are as good as the original particles. 4And this is precisely how Dirac predicted the existence of the positron (in 1928). 5And that is how physics community was thinking of neutrinos, before discovering that the mass of the neutrino is ﬁnite.

3 Indeed, " k − E m # det = E2 − k2 − m2. m −k − E We thus conclude that the massive theory is supposed to deal with a hybridization between two Weyl states, one state having the energy +k and the other state having the energy −k. Rotational symmetry implies that the hybrid state should have a well-deﬁned helicity. This leaves us with only one option, which is the hybridization between the two bi-spinor states of the same handedness and opposite energy. The Hamiltonian is obtained from (7) by adding the mixing terms:

" H(+) mI # H = . (8) mI H(−) Here I is the identity 2 × 2 matrix. The resulting differential equation is called Dirac equation. In view of its matrix form, Dirac Hamiltonian features different representations. Two of them are especially important. The one we just introduced is convenient for the ultra-relativistic limit k m. Here, in the momentum representation, we have " ~σ·k mI # H = , (9) k mI −~σ·k and see that the off-diagonal mass terms play the role of small perturbation. With the unitary transformation 1 " 1 1 #" ~σ·k mI #" 1 1 # " mI ~σ·k # = . 2 1 −1 mI −~σ·k 1 −1 ~σ·k −mI we get the representation " mI ~σ·k # H = , (10) k ~σ·k −mI which is most convenient in the non-relativistic limit k m. Here the kinetic term ~σ · k is a small (mixing) perturbation on top of the diagonal rest-energy term. For example, the non-relativistic energy can be found by the second-order perturbation theory: k2 k2 E ≈ m + = m + . k m − (−m) 2m

4 In the coordinate representation and the representation (10) for the matrix part of the Hamiltonian, the Dirac equation for the 4-component bi- spinor |Φi reads (now we restoreh ¯ and c)

∂ ih¯ |Φi = (−ihc¯ ~α·∇ + mc2β) |Φi, ∂t where " 0 ~σ # " I 0 # ~α = , β = . ~σ 0 0 −I